What's the Expected Value of number of runs required for a hellfire ring?
On Monster Power 1, the drop rate for each key 10%. If I do 10 runs, the chance of having one key dropped in these 10 runs is 1-(1-10%)^10, or about 65%. After getting 3 keys, I can try to get a loot from a uber with a chance of 10%. I will need 3 loot from uber bosses to craft one ring. How do I calculate the Expected Value of total number of runs I will need?
Edit: from wikipedia, I just learned the Mean of a Geometric Distribution is 1/p. So the Expected Value of getting a key drop on Monster Power 1 should be 10 runs. But then, if 10 is the mean, why would 1-(1-10%)^10 ~= 65%?
Solution 1:
Your formula is correct: if the chances of getting a ring in one run are 1/10, then the expected number of runs is 10. And if that is the case, then the expected number of runs to get 3 rings is 3*10=30 runs.
Once you have all three keys, you craft them into a portal, which you use to get a chance at one of three organs. Assuming these organs also have a 10% drop rate, that means you'll need an average of 10 portals ( = 30 keys = 300 key-runs) per organ. Since you need 3 organs, your overall expected number of runs is 900 for all the keys you need, plus another 30 to get the organs, for a total of 930 (at MP1).
In general, if k is the probability of a key dropping and o is the probability of an organ dropping, your expected number of runs (including both keys and organs) will be 9/po + 3/o.
Note that the error in your logic is that you appear to be assuming the expected value is the same as the point at which you have 50% chance of succeeding - it's not. Take coin-flips for example; you have a 50% chance of getting heads after only one flip, but the expected number of flips to see a heads is 2.
Solution 2:
fbueckert's table still isn't quite right. Here's a fixed version:
+----+-------+----------+---------+--------+
| MP | 1 key | 1 portal | 1 organ | 1 ring |
+----+-------+----------+---------+--------+
| 0 | 20 | 60 | 1200 | 3600 |
| 1 | 10 | 30 | 300 | 900 |
| 2 | 5 | 15 | 75 | 225 |
| 3 | 4 | 10 | 34 | 100 |
| 4 | 3 | 8 | 19 | 57 |
| 5 | 2 | 6 | 12 | 36 |
| 6 | 2 | 5 | 9 | 25 |
| 7 | 2 | 5 | 7 | 19 |
| 8 | 2 | 4 | 5 | 15 |
| 9 | 2 | 4 | 4 | 12 |
| 10 | 1 | 3 | 3 | 9 |
+----+-------+----------+---------+--------+
It looks like MP3 and MP4 never got the previous fix, while MP6 and MP7 were rounded down instead of up (except in the first column). Here's my work if anyone wants to check it:
+----+-------+----------+---------+--------+
| MP | 1 key | 1 portal | 1 organ | 1 ring |
+----+-------+----------+---------+--------+
| 0 | 20 | 60 | 1200 | 3600 |
| 1 | 10 | 30 | 300 | 900 |
| 2 | 5 | 15 | 75 | 225 |
| 3 | 10/3 | 10 | 100/3 | 100 |
| 4 | 5/2 | 15/2 | 75/4 | 225/4 |
| 5 | 2 | 6 | 12 | 36 |
| 6 | 5/3 | 5 | 25/3 | 25 |
| 7 | 10/7 | 30/7 | 300/49 | 900/49 |
| 8 | 5/4 | 15/4 | 75/16 | 225/16 |
| 9 | 10/9 | 10/3 | 100/27 | 100/9 |
| 10 | 1 | 3 | 3 | 9 |
+----+-------+----------+---------+--------+
Sorry about answering instead of commenting. Apparently I need reputation to comment but not to answer. The MP6/7 differences are minor, but the MP3/4 differences are significant.
Solution 3:
As everyone pointed out, there is no way to guarantee a specific key drop. It gets pretty close, but it never quite gets there.
For expected value, it's just adding the chances. 10% chance of a drop means the expected value is 10. 20% means you the expected value is 5, etc.
Monster Power 1 key 1 portal 1 organ 1 ring
---------------------------------------------------------
No Monster Power (5%): 20 60 1,200 3,600
MP 1 10 30 300 900
MP 2 5 15 75 225
MP 3 4 12 48 144
MP 4 3 9 27 81
MP 5 2 6 12 36
MP 6 2 5 8 24
MP 7 2 4 6 18
MP 8 2 4 5 15
MP 9 2 4 4 12
MP 10 1 3 3 9
This makes the assumption that you do all your runs at the exact same MP. I rounded up to the nearest whole value.